Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 12.6s

Alternatives: 8

Speedup: N/A×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t_0\\ \frac{t_0}{\left(s \cdot t_1\right) \cdot t_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t_0\\ \frac{t_0}{\left(s \cdot t_1\right) \cdot t_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Alternative 1: 99.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s \cdot e^{\frac{-x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (/ x s))) (+ s (* s (exp (/ (- x) s)))))))

C

float code(float x, float s) {
	return 1.0f / ((1.0f + expf((x / s))) * (s + (s * expf((-x / s)))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((1.0e0 + exp((x / s))) * (s + (s * exp((-x / s)))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + exp(Float32(x / s))) * Float32(s + Float32(s * exp(Float32(Float32(-x) / s))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((single(1.0) + exp((x / s))) * (s + (s * exp((-x / s)))));
end

Derivation

1. Initial program 99.4%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}}\right)} \]

   2. sqrt-unprod 94.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{s}}\right)} \]

   3. sqr-neg 94.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}}\right)} \]

   4. sqrt-unprod -0.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}}\right)} \]

   5. add-sqr-sqrt 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{-\left|x\right|}}{s}}\right)} \]

   6. expm1-log1p-u 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   7. expm1-udef 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)} - 1\right)}} \]

6. Applied egg-rr 65.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)} - 1\right)}} \]
7. Step-by-step derivation

   1. expm1-def 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)\right)}} \]

   2. expm1-log1p 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\frac{x}{s}} + 1\right)}} \]

   3. +-commutative 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

8. Simplified 65.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]
9. Step-by-step derivation

   1. distribute-frac-neg 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\color{blue}{-\frac{\left|x\right|}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   2. rec-exp 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   3. add-sqr-sqrt 55.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   4. fabs-sqr 55.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   5. add-sqr-sqrt 99.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\color{blue}{x}}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

10. Applied egg-rr 99.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{e^{\frac{x}{s}}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]
11. Step-by-step derivation

    1. rec-exp 99.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{e^{-\frac{x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

    2. distribute-neg-frac 99.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\color{blue}{\frac{-x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

12. Simplified 99.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{e^{\frac{-x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

13. Taylor expanded in x around inf 99.5%

    \[\leadsto \frac{1}{\color{blue}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s \cdot e^{-1 \cdot \frac{x}{s}}\right)}} \]
14. Step-by-step derivation

    1. mul-1-neg 99.5%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s \cdot e^{\color{blue}{-\frac{x}{s}}}\right)} \]

    2. exp-neg 99.5%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s \cdot \color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right)} \]

15. Applied egg-rr 99.5%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s \cdot \color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right)} \]

17. Simplified 99.5%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s \cdot \color{blue}{e^{\frac{-x}{s}}}\right)} \]

18. Final simplification 99.5%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s \cdot e^{\frac{-x}{s}}\right)} \]
19. Add Preprocessing

Alternative 2: 99.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(\left(1 + e^{\frac{x}{s}}\right) \cdot \left(1 + e^{\frac{-x}{s}}\right)\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (* (+ 1.0 (exp (/ x s))) (+ 1.0 (exp (/ (- x) s)))))))

C

float code(float x, float s) {
	return 1.0f / (s * ((1.0f + expf((x / s))) * (1.0f + expf((-x / s)))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * ((1.0e0 + exp((x / s))) * (1.0e0 + exp((-x / s)))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(Float32(1.0) + exp(Float32(x / s))) * Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * ((single(1.0) + exp((x / s))) * (single(1.0) + exp((-x / s)))));
end

Derivation

1. Initial program 99.4%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}}\right)} \]

   2. sqrt-unprod 94.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{s}}\right)} \]

   3. sqr-neg 94.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}}\right)} \]

   4. sqrt-unprod -0.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}}\right)} \]

   5. add-sqr-sqrt 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{-\left|x\right|}}{s}}\right)} \]

   6. expm1-log1p-u 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   7. expm1-udef 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)} - 1\right)}} \]

6. Applied egg-rr 65.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)} - 1\right)}} \]
7. Step-by-step derivation

   1. expm1-def 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)\right)}} \]

   2. expm1-log1p 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\frac{x}{s}} + 1\right)}} \]

   3. +-commutative 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

8. Simplified 65.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]
9. Step-by-step derivation

   1. distribute-frac-neg 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\color{blue}{-\frac{\left|x\right|}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   2. rec-exp 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   3. add-sqr-sqrt 55.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   4. fabs-sqr 55.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   5. add-sqr-sqrt 99.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\color{blue}{x}}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

10. Applied egg-rr 99.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{e^{\frac{x}{s}}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]
11. Step-by-step derivation

    1. rec-exp 99.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{e^{-\frac{x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

    2. distribute-neg-frac 99.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\color{blue}{\frac{-x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

12. Simplified 99.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{e^{\frac{-x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

13. Taylor expanded in s around 0 99.5%

    \[\leadsto \frac{1}{\color{blue}{s \cdot \left(\left(1 + e^{-1 \cdot \frac{x}{s}}\right) \cdot \left(1 + e^{\frac{x}{s}}\right)\right)}} \]
14. Step-by-step derivation

    1. mul-1-neg 99.5%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s \cdot e^{\color{blue}{-\frac{x}{s}}}\right)} \]

    2. exp-neg 99.5%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s \cdot \color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right)} \]

15. Applied egg-rr 99.5%

    \[\leadsto \frac{1}{s \cdot \left(\left(1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}\right) \cdot \left(1 + e^{\frac{x}{s}}\right)\right)} \]

17. Simplified 99.5%

    \[\leadsto \frac{1}{s \cdot \left(\left(1 + \color{blue}{e^{\frac{-x}{s}}}\right) \cdot \left(1 + e^{\frac{x}{s}}\right)\right)} \]

18. Final simplification 99.5%

    \[\leadsto \frac{1}{s \cdot \left(\left(1 + e^{\frac{x}{s}}\right) \cdot \left(1 + e^{\frac{-x}{s}}\right)\right)} \]
19. Add Preprocessing

Alternative 3: 60.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ 1.0 (exp (/ x s))))))

C

float code(float x, float s) {
	return 0.5f / (s * (1.0f + expf((x / s))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (s * (1.0e0 + exp((x / s))))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(s * Float32(Float32(1.0) + exp(Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (s * (single(1.0) + exp((x / s))));
end

Derivation

1. Initial program 99.4%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}}\right)} \]

   2. sqrt-unprod 94.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{s}}\right)} \]

   3. sqr-neg 94.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}}\right)} \]

   4. sqrt-unprod -0.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}}\right)} \]

   5. add-sqr-sqrt 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{-\left|x\right|}}{s}}\right)} \]

   6. expm1-log1p-u 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   7. expm1-udef 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)} - 1\right)}} \]

6. Applied egg-rr 65.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)} - 1\right)}} \]
7. Step-by-step derivation

   1. expm1-def 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)\right)}} \]

   2. expm1-log1p 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\frac{x}{s}} + 1\right)}} \]

   3. +-commutative 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

8. Simplified 65.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

9. Taylor expanded in s around inf 62.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{1}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

10. Taylor expanded in s around 0 62.3%

    \[\leadsto \color{blue}{\frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)}} \]

11. Final simplification 62.3%

    \[\leadsto \frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \]
12. Add Preprocessing

Alternative 4: 76.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999936531045e-21)
   (/ 1.0 (+ (* s 4.0) (* x (/ x s))))
   (/ 1.0 (exp (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999936531045e-21f) {
		tmp = 1.0f / ((s * 4.0f) + (x * (x / s)));
	} else {
		tmp = 1.0f / expf((x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.999999936531045e-21) then
        tmp = 1.0e0 / ((s * 4.0e0) + (x * (x / s)))
    else
        tmp = 1.0e0 / exp((x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999936531045e-21))
		tmp = Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / exp(Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999936531045e-21))
		tmp = single(1.0) / ((s * single(4.0)) + (x * (x / s)));
	else
		tmp = single(1.0) / exp((x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999e-21

   1. Initial program 99.1%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}}\right)} \]

      2. sqrt-unprod 89.8%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{s}}\right)} \]

      3. sqr-neg 89.8%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}}\right)} \]

      4. sqrt-unprod -0.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}}\right)} \]

      5. add-sqr-sqrt 34.4%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{-\left|x\right|}}{s}}\right)} \]

      6. expm1-log1p-u 34.4%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      7. expm1-udef 34.4%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)} - 1\right)}} \]

   6. Applied egg-rr 40.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)} - 1\right)}} \]
   7. Step-by-step derivation

      1. expm1-def 40.4%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)\right)}} \]

      2. expm1-log1p 40.4%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\frac{x}{s}} + 1\right)}} \]

      3. +-commutative 40.4%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

   8. Simplified 40.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]
   9. Step-by-step derivation

      1. distribute-frac-neg 40.4%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\color{blue}{-\frac{\left|x\right|}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

      2. rec-exp 40.5%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

      3. add-sqr-sqrt 24.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

      4. fabs-sqr 24.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

      5. add-sqr-sqrt 99.2%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\color{blue}{x}}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   10. Applied egg-rr 99.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{e^{\frac{x}{s}}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]
   11. Step-by-step derivation

       1. rec-exp 99.2%

          \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{e^{-\frac{x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

       2. distribute-neg-frac 99.2%

          \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\color{blue}{\frac{-x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   12. Simplified 99.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{e^{\frac{-x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   13. Taylor expanded in x around 0 70.7%

       \[\leadsto \frac{1}{\color{blue}{4 \cdot s + \frac{{x}^{2}}{s}}} \]
   14. Step-by-step derivation

       1. div-inv 70.7%

          \[\leadsto \frac{1}{4 \cdot s + \color{blue}{{x}^{2} \cdot \frac{1}{s}}} \]

       2. unpow2 70.7%

          \[\leadsto \frac{1}{4 \cdot s + \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{s}} \]

       3. associate-*l* 71.7%

          \[\leadsto \frac{1}{4 \cdot s + \color{blue}{x \cdot \left(x \cdot \frac{1}{s}\right)}} \]

       4. div-inv 71.7%

          \[\leadsto \frac{1}{4 \cdot s + x \cdot \color{blue}{\frac{x}{s}}} \]

   15. Applied egg-rr 71.7%

       \[\leadsto \frac{1}{4 \cdot s + \color{blue}{x \cdot \frac{x}{s}}} \]

   if 1.9999999e-21 < x

   1. Initial program 99.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}}\right)} \]

      2. sqrt-unprod 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{s}}\right)} \]

      3. sqr-neg 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}}\right)} \]

      4. sqrt-unprod -0.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}}\right)} \]

      5. add-sqr-sqrt 9.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{-\left|x\right|}}{s}}\right)} \]

      6. expm1-log1p-u 9.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      7. expm1-udef 9.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)} - 1\right)}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)} - 1\right)}} \]
   7. Step-by-step derivation

      1. expm1-def 99.8%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)\right)}} \]

      2. expm1-log1p 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\frac{x}{s}} + 1\right)}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

   8. Simplified 99.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

   9. Taylor expanded in s around inf 96.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{1}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]
   10. Step-by-step derivation

       1. /-rgt-identity 96.8%

          \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(s, 1, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)}{1}}} \]

       2. clear-num 96.8%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(s, 1, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)}}}} \]

       3. associate-/r* 95.9%

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\frac{1}{\mathsf{fma}\left(s, 1, s\right)}}{1 + e^{\frac{x}{s}}}}}} \]

       4. fma-udef 95.9%

          \[\leadsto \frac{1}{\frac{1}{\frac{\frac{1}{\color{blue}{s \cdot 1 + s}}}{1 + e^{\frac{x}{s}}}}} \]

       5. *-rgt-identity 95.9%

          \[\leadsto \frac{1}{\frac{1}{\frac{\frac{1}{\color{blue}{s} + s}}{1 + e^{\frac{x}{s}}}}} \]

   11. Applied egg-rr 95.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\frac{1}{s + s}}{1 + e^{\frac{x}{s}}}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 95.9%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{1}{s + s}} \cdot \left(1 + e^{\frac{x}{s}}\right)}} \]

       2. remove-double-div 96.8%

          \[\leadsto \frac{1}{\color{blue}{\left(s + s\right)} \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

       3. distribute-lft-in 96.8%

          \[\leadsto \frac{1}{\color{blue}{\left(s + s\right) \cdot 1 + \left(s + s\right) \cdot e^{\frac{x}{s}}}} \]

       4. metadata-eval 96.8%

          \[\leadsto \frac{1}{\left(s + s\right) \cdot \color{blue}{\frac{1}{1}} + \left(s + s\right) \cdot e^{\frac{x}{s}}} \]

       5. div-inv 96.8%

          \[\leadsto \frac{1}{\color{blue}{\frac{s + s}{1}} + \left(s + s\right) \cdot e^{\frac{x}{s}}} \]

       6. /-rgt-identity 96.8%

          \[\leadsto \frac{1}{\color{blue}{\left(s + s\right)} + \left(s + s\right) \cdot e^{\frac{x}{s}}} \]

       7. flip-+ -0.0%

          \[\leadsto \frac{1}{\color{blue}{\frac{s \cdot s - s \cdot s}{s - s}} + \left(s + s\right) \cdot e^{\frac{x}{s}}} \]

       8. +-inverses -0.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{0}}{s - s} + \left(s + s\right) \cdot e^{\frac{x}{s}}} \]

       9. +-inverses -0.0%

          \[\leadsto \frac{1}{\frac{0}{\color{blue}{0}} + \left(s + s\right) \cdot e^{\frac{x}{s}}} \]

       10. flip-+ -0.0%

           \[\leadsto \frac{1}{\frac{0}{0} + \color{blue}{\frac{s \cdot s - s \cdot s}{s - s}} \cdot e^{\frac{x}{s}}} \]

       11. +-inverses -0.0%

           \[\leadsto \frac{1}{\frac{0}{0} + \frac{\color{blue}{0}}{s - s} \cdot e^{\frac{x}{s}}} \]

       12. +-inverses -0.0%

           \[\leadsto \frac{1}{\frac{0}{0} + \frac{0}{\color{blue}{0}} \cdot e^{\frac{x}{s}}} \]

   13. Applied egg-rr -0.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{0}{0} + \frac{0}{0} \cdot e^{\frac{x}{s}}}} \]

   15. Simplified 93.0%

       \[\leadsto \frac{1}{\color{blue}{0 + e^{\frac{x}{s}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.1% accurate, 56.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot 4 + x \cdot \frac{x}{s}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ (* s 4.0) (* x (/ x s)))))

C

float code(float x, float s) {
	return 1.0f / ((s * 4.0f) + (x * (x / s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((s * 4.0e0) + (x * (x / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(x / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((s * single(4.0)) + (x * (x / s)));
end

Derivation

1. Initial program 99.4%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}}\right)} \]

   2. sqrt-unprod 94.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{s}}\right)} \]

   3. sqr-neg 94.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}}\right)} \]

   4. sqrt-unprod -0.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}}\right)} \]

   5. add-sqr-sqrt 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{-\left|x\right|}}{s}}\right)} \]

   6. expm1-log1p-u 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   7. expm1-udef 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)} - 1\right)}} \]

6. Applied egg-rr 65.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)} - 1\right)}} \]
7. Step-by-step derivation

   1. expm1-def 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)\right)}} \]

   2. expm1-log1p 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\frac{x}{s}} + 1\right)}} \]

   3. +-commutative 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

8. Simplified 65.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]
9. Step-by-step derivation

   1. distribute-frac-neg 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\color{blue}{-\frac{\left|x\right|}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   2. rec-exp 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   3. add-sqr-sqrt 55.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   4. fabs-sqr 55.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   5. add-sqr-sqrt 99.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\color{blue}{x}}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

10. Applied egg-rr 99.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{e^{\frac{x}{s}}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]
11. Step-by-step derivation

    1. rec-exp 99.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{e^{-\frac{x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

    2. distribute-neg-frac 99.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\color{blue}{\frac{-x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

12. Simplified 99.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{e^{\frac{-x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

13. Taylor expanded in x around 0 63.9%

    \[\leadsto \frac{1}{\color{blue}{4 \cdot s + \frac{{x}^{2}}{s}}} \]
14. Step-by-step derivation

    1. div-inv 63.9%

       \[\leadsto \frac{1}{4 \cdot s + \color{blue}{{x}^{2} \cdot \frac{1}{s}}} \]

    2. unpow2 63.9%

       \[\leadsto \frac{1}{4 \cdot s + \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{s}} \]

    3. associate-*l* 64.5%

       \[\leadsto \frac{1}{4 \cdot s + \color{blue}{x \cdot \left(x \cdot \frac{1}{s}\right)}} \]

    4. div-inv 64.5%

       \[\leadsto \frac{1}{4 \cdot s + x \cdot \color{blue}{\frac{x}{s}}} \]

15. Applied egg-rr 64.5%

    \[\leadsto \frac{1}{4 \cdot s + \color{blue}{x \cdot \frac{x}{s}}} \]

16. Final simplification 64.5%

    \[\leadsto \frac{1}{s \cdot 4 + x \cdot \frac{x}{s}} \]
17. Add Preprocessing

Alternative 6: 51.1% accurate, 68.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(\frac{x}{s} + 4\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* s (+ (/ x s) 4.0))))

C

float code(float x, float s) {
	return 1.0f / (s * ((x / s) + 4.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * ((x / s) + 4.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(x / s) + Float32(4.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * ((x / s) + single(4.0)));
end

Derivation

1. Initial program 99.4%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}}\right)} \]

   2. sqrt-unprod 94.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{s}}\right)} \]

   3. sqr-neg 94.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}}\right)} \]

   4. sqrt-unprod -0.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}}\right)} \]

   5. add-sqr-sqrt 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{-\left|x\right|}}{s}}\right)} \]

   6. expm1-log1p-u 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   7. expm1-udef 23.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)} - 1\right)}} \]

6. Applied egg-rr 65.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)} - 1\right)}} \]
7. Step-by-step derivation

   1. expm1-def 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)\right)}} \]

   2. expm1-log1p 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\frac{x}{s}} + 1\right)}} \]

   3. +-commutative 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

8. Simplified 65.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]
9. Step-by-step derivation

   1. distribute-frac-neg 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\color{blue}{-\frac{\left|x\right|}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   2. rec-exp 65.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   3. add-sqr-sqrt 55.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   4. fabs-sqr 55.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

   5. add-sqr-sqrt 99.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, \frac{1}{e^{\frac{\color{blue}{x}}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

10. Applied egg-rr 99.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{e^{\frac{x}{s}}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]
11. Step-by-step derivation

    1. rec-exp 99.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{e^{-\frac{x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

    2. distribute-neg-frac 99.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\color{blue}{\frac{-x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

12. Simplified 99.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(s, \color{blue}{e^{\frac{-x}{s}}}, s\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

13. Taylor expanded in s around 0 99.5%

    \[\leadsto \frac{1}{\color{blue}{s \cdot \left(\left(1 + e^{-1 \cdot \frac{x}{s}}\right) \cdot \left(1 + e^{\frac{x}{s}}\right)\right)}} \]

14. Taylor expanded in s around inf 24.9%

    \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 + \left(-2 \cdot \frac{x}{s} + 2 \cdot \frac{x}{s}\right)\right)}} \]

16. Simplified 49.1%

    \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 + \frac{x}{s}\right)}} \]

17. Final simplification 49.1%

    \[\leadsto \frac{1}{s \cdot \left(\frac{x}{s} + 4\right)} \]
18. Add Preprocessing

Alternative 7: 29.2% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.006000000052154064:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.006000000052154064) (/ 0.25 s) (/ 1.0 (* x 2.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.006000000052154064f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x * 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.006000000052154064e0) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.006000000052154064))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.006000000052154064))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00600000005

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      2. distribute-lft-in 99.3%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \color{blue}{\left(s \cdot 1 + s \cdot e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. *-rgt-identity 99.3%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(\color{blue}{s} + s \cdot e^{\frac{-\left|x\right|}{s}}\right)} \]

      4. fabs-neg 99.3%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + s \cdot e^{\frac{-\color{blue}{\left|-x\right|}}{s}}\right)} \]

      5. distribute-frac-neg 99.3%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + s \cdot e^{\color{blue}{-\frac{\left|-x\right|}{s}}}\right)} \]

      6. exp-neg 99.2%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + s \cdot \color{blue}{\frac{1}{e^{\frac{\left|-x\right|}{s}}}}\right)} \]

      7. associate-*r/ 99.2%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \color{blue}{\frac{s \cdot 1}{e^{\frac{\left|-x\right|}{s}}}}\right)} \]

      8. *-rgt-identity 99.2%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{\color{blue}{s}}{e^{\frac{\left|-x\right|}{s}}}\right)} \]

      9. *-lft-identity 99.2%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\color{blue}{1 \cdot \frac{\left|-x\right|}{s}}}}\right)} \]

      10. metadata-eval 99.2%

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\color{blue}{\frac{-1}{-1}} \cdot \frac{\left|-x\right|}{s}}}\right)} \]

      11. times-frac 99.2%

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\color{blue}{\frac{-1 \cdot \left|-x\right|}{-1 \cdot s}}}}\right)} \]

      12. neg-mul-1 99.2%

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{-1 \cdot \left|-x\right|}{\color{blue}{-s}}}}\right)} \]

      13. neg-mul-1 99.2%

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{\color{blue}{-\left|-x\right|}}{-s}}}\right)} \]

      14. fabs-neg 99.2%

          \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{-\color{blue}{\left|x\right|}}{-s}}}\right)} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf 34.0%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   if 0.00600000005 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}}\right)} \]

      2. sqrt-unprod 100.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{s}}\right)} \]

      3. sqr-neg 100.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}}\right)} \]

      4. sqrt-unprod -0.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}}\right)} \]

      5. add-sqr-sqrt 4.5%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \left(1 + e^{\frac{\color{blue}{-\left|x\right|}}{s}}\right)} \]

      6. expm1-log1p-u 4.5%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      7. expm1-udef 4.5%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)} - 1\right)}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)} - 1\right)}} \]
   7. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)\right)}} \]

      2. expm1-log1p 100.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\frac{x}{s}} + 1\right)}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(1 + e^{\frac{x}{s}}\right)}} \]

   9. Taylor expanded in s around inf 3.0%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \left|x\right| + \left(2 \cdot x + 4 \cdot s\right)}} \]

   10. Taylor expanded in x around inf 14.4%

       \[\leadsto \frac{1}{\color{blue}{2 \cdot x}} \]
   11. Step-by-step derivation

       1. *-commutative 14.4%

          \[\leadsto \frac{1}{\color{blue}{x \cdot 2}} \]

   12. Simplified 14.4%

       \[\leadsto \frac{1}{\color{blue}{x \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.006000000052154064:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.5% accurate, 206.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

float code(float x, float s) {
	return 0.25f / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / s;
end

Derivation

1. Initial program 99.4%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   2. distribute-lft-in 99.5%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \color{blue}{\left(s \cdot 1 + s \cdot e^{\frac{-\left|x\right|}{s}}\right)}} \]

   3. *-rgt-identity 99.5%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(\color{blue}{s} + s \cdot e^{\frac{-\left|x\right|}{s}}\right)} \]

   4. fabs-neg 99.5%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + s \cdot e^{\frac{-\color{blue}{\left|-x\right|}}{s}}\right)} \]

   5. distribute-frac-neg 99.5%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + s \cdot e^{\color{blue}{-\frac{\left|-x\right|}{s}}}\right)} \]

   6. exp-neg 99.4%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + s \cdot \color{blue}{\frac{1}{e^{\frac{\left|-x\right|}{s}}}}\right)} \]

   7. associate-*r/ 99.4%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \color{blue}{\frac{s \cdot 1}{e^{\frac{\left|-x\right|}{s}}}}\right)} \]

   8. *-rgt-identity 99.4%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{\color{blue}{s}}{e^{\frac{\left|-x\right|}{s}}}\right)} \]

   9. *-lft-identity 99.4%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\color{blue}{1 \cdot \frac{\left|-x\right|}{s}}}}\right)} \]

   10. metadata-eval 99.4%

       \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\color{blue}{\frac{-1}{-1}} \cdot \frac{\left|-x\right|}{s}}}\right)} \]

   11. times-frac 99.4%

       \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\color{blue}{\frac{-1 \cdot \left|-x\right|}{-1 \cdot s}}}}\right)} \]

   12. neg-mul-1 99.4%

       \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{-1 \cdot \left|-x\right|}{\color{blue}{-s}}}}\right)} \]

   13. neg-mul-1 99.4%

       \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{\color{blue}{-\left|-x\right|}}{-s}}}\right)} \]

   14. fabs-neg 99.4%

       \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{-\color{blue}{\left|x\right|}}{-s}}}\right)} \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 25.9%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]

6. Final simplification 25.9%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))